1.6

What if it is Reflected More than Once?

Pg. 23Rigid Transformations: Translations

1.6 – What if it is reflected more than Once?_____Rigid Transformations: Translations

In Lesson 1.5, you learned how to change a shape by reflecting it across a line, like the ice cream cones shown at right. Today you will learn more about reflections and learn about a new type of transformation: translations.

1.38 – TWO REFLECTIONS As Amanda was finding reflections, she wondered, “What if I reflect a shape twice over parallel lines?” Investigate her question as you answer the questions below.

a. Find ∆ABC and lines n and p (shown below). What happens when ∆ABC isreflected across line n to form and then is reflected across line p to form First visualize the reflections and then test your idea of the result by drawing both reflections.

Prediction

Drawing

'A

'B 'C

''A

''B''C

b. Examine your result from part (a). Compare the original triangle ∆ABC with the final result, What single motion would change ∆ABC to

'' '' ''.A B C'' '' ''?A B C

Moving it over, sliding

Geogebra Reflections

c. Amanda analyzed her results from part (a). “It Just looks like I could have just slid ∆ABC over!” Sliding a shape from its original position to a new position is called translating. For example, the ice cream cone at right has been translated. Notice that the image of the ice cream cone has the same orientation as the original (that is, it is not turned or flipped). What words can you use to describe a translation?

Moving it over, sliding

Translation

Transformation

 

Moving the shape in some way

Sliding shape over

Translation

Slid over, not flipped

Right 7

Down 3

Motion Rule:

( , )x y

Rightor

Left

Upor

Down

#, x #y

Right 7

Down 3

x, y ( 7,x y 3)

Left 4 Up 1 Right 3 Down 7

Down 5 Right 8

x, y

+4

–3

(x 4, y 3)

x, y

–5

+6

(x 5, y 6)

x, y

–8

–7

(x 8, y 7)

'EF

G

E 'F

'G

(2, 3)

(-1, 5)

(2, -1)

AB

C'A B '

C '

(-2, -2)

(2, -1)

(4, -5)

f. Can you find the new point without counting on the graph? Use the motion rule to find if P is at (2, -1).

(2 – 3, -1 + 1)

' 1,0P

(2 + 7, -1 – 3)

' 9, 4P

(2 + 5, -1)

' 7, 1P

1.40 – NON-CONGRUENT RULES Use the following rules to find the new shape by plugging in each x and y value to find the new coordinate.

(-1, -4) (0, -2) (3, -4)

'A

'B

'C

(-6, -4) (-4, -2) (2, -4)

'A

'B

'C

c. What is the difference between (a) and (b)? Why do you think one is congruent to the original and one is not?

'A

'B

'C'A

'B

'CMultiplying changes the size of the shape

1.41 – WORKING BACKWARDS What if you are only given the location of the translated shape? Can you find the original shape?

'X

'Z

'Y

'X

Right 4

Down 1

Left 4

Up 1

X ' Y '

Z '

X

Y

Z(-2, -2)

(0, -4)

(1, 2)

A '

C ' B '

Left 3

Right 3

A

B C

A '

C ' B '

A

B C

(6, -1)

(6, -4)

(4, -4)